By J. Garnett
Ebook through Garnett, J.
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Additional info for Analytic Capacity and Measure
1( z ) dxdy "0-; Since by Green's theorem o ~ g(O 1 ~dxdy (j-; '" we get (1. 3) via. Fubini's theorem. 3: If ~ on an open set V, then Proof: is almost everywhere equal to a function analytic I~I (V) = O. 1. , then C: ~ ~ o. in the compactly supported continuous functions. 5) and the obviously necessary conditions actually determine the Cauchy transform ~ of ~. ",f(z) = 0 allu let iJ. be a. -39- compactly supported measure such that fez} Then Proof: ~ (If iJ-; 0 almost everywhere. ~ ~(z) Replacing f weakly, and we must show xp Izl < pl.
Z ) so that F = g almost everywhere. L~oc be analytiC off a compact set E and Il be a measure on E. e . There is an admissible grid JfdR If(z) 0, F ... t ~l(R ) -42- for all Proof: JilR R ~ 6'/,. Assume (1) holds. 1. Now and assume (ii) holds. t this implies R. 2. nd analyt ic on all curves 00. t ~ J for all Then fez) - Q(z). = D\E. D\,E. 2. The set of in D. 2 extends analytically to D. This result is extended somewhat in the next section. kened. We now describe these f I fELIce which are almost everywhere -43- Cauchy transforms.
Hence the F. and M. Izl=r I(z h(Z) ~ qJ'(z)/21r1. for h with k > 0 and by HI. We can € 0 < r < I, and ~(z)h(z)dz = 0 • )